Limit Question Solution | Simplifying Infinite Limit with e^2 Result

[talolard]

Homework Statement

I've done thsi limit, i think I'm half right. I'm sure I have done something wrong, but I'm not sure what.

$$lim_{x-> \infty} ( \frac {x+lnx}{x-lnx})^(x/lnx)$$

The Attempt at a Solution

$$lim_{x-> \infty} ( \frac {x+lnx}{x-lnx})^{x/lnx} = lim_{x-> \infty} ( \frac {x(1 +\frac {lnx}{x})}{x(1- \frac {lnx}{x})})^{x/lnx}=$$
$$t= \frac {x}{lnx}; lim_{t-> \infty} (\frac {1 + \frac {1}{t}} {1 - \frac{1}{t}})^t=$$
$$\frac {e}{e^{-1}}=e^2$$

 

[vela]

That's right.